COMPARING TWO DIFFERENT CONTROL ALGORITHMS APPLIED TO DYNAMIC POSITIONING OF A PIPELINE LAUNCHING BARGE

Transcription

1 Proceedings of the th Mediterranean Conference on Control and Automation - MED Lisbon, Portugal, July 9-,. COMPAING TWO DIFFEENT CONTOL ALGOITHMS APPLIED TO DYNAMIC POSITIONING OF A PIPELINE LAUNCHING BAGE E.A. Tannuri*, C.P. Pesce Dept. of Mechanical Engineering, Escola Politécnica, University of São Paulo São Paulo, Brazil, Av. Prof. Mello Moraes, SP fax: *eduat@usp.br ; ceppesce@usp.br Keywords: sliding mode control, optimal control, dynamic positioning, launching barge. Abstract This work was motivated by the necessity of estimating the required power for a Dynamic Positioning System (DPS) to be installed in an already existing pipeline-laying barge, which operates in both intermediate and deep waters (up to meters). Small-scale experiments were used to obtain current and wind forces acting during operation. Additional environmental effects were estimated using validated models. A numerical simulator was then developed, also including propeller dynamics, thruster allocation logics and the controller. A robust non-linear sliding mode control (SM) was applied, and was compared to the so-common optimal linear LQ control. This paper describes, in details, the application of both controllers and the methodology used for comparison, focusing on: dynamic performance and energy consumption, number of adjusting parameters, implementation simplicity and robustness to modelling errors. The analyses showed that both controllers satisfy operational performance requirements, although the SM controller is more appropriate, due to good robustness properties and fewer number of parameters to be adjusted. The LQ controller is simpler, but extremely dependent on weight matrix adjustment, what is very time-consuming and must be redone whenever the barge heading changes. Introduction BGL is a crane and pipe-laying barge (Figure ), operating in Brazilian waters for more than years. Equipped with mooring lines and operated with the aid of anchor handling tugboats, BGL was originally designed for shallow and intermediate depths. Pipe-laying has been successfully accomplished through the so-called S- Lay operation, shown in Figure. BGL S-Lay is done from a stern launching ramp, positioned at starboard, with or without a stinger. Pipeline is welded on deck.traction is sustained by means of a traction-controlling machine, of the caterpillar type. Figure : BGL photography With increasing depths, the based on conventional mooring positioning laying operation presents serious technical and economical limitations. S-Lay mode of operation is appropriate for shallow and intermediate depths. For deeper waters the socalled J-Lay launching is a recommended practice (Figure ), in which the pipe is launched almost vertically.

2 The operation without DP system requires an extremely time consuming planning of anchor positioning along all the launching path. Furthermore, the complex operation usually comprises 5 anchor handling tugboats and several operators to control the winch pull-in machines. Figure : S-lay launching operation Figure : J-lay launching operation A robust DP system can then improve stationkeeping ability, for both S- or J-Laying modes, with no loss of safety, enhancing operational time schedule and making the operation economically feasible. A previous work described in details the procedure used to determine the required power for the DPS [], guaranteeing safe operation in deep-water even with simultaneous failure of thrusters. Environmental forces were evaluated with towing tank tests complemented by validated theoretical models. A numerical simulator was then developed and an ideal feedback linearization controller was used in the exhaustive dynamic analysis then worked out. As a natural extension, this work deals with the practical implementation of the controller, taking into account robustness issues, performance under several environmental conditions, design simplicity and parameter adjusting procedures. Two different controllers were implemented and tested, both presenting very different design philosophies and representing two broad classes of DPS controllers usually applied. The LQ controller takes a linearized model of barge dynamics and all environmental forces are considered as disturbances applied to the control loop. With proper adjustment of matrix weights, the controller can be tuned in order to guarantee prescribed performance parameters under a range of disturbance amplitudes. This kind of controller is a natural extension of the so common uncoupled PID control, taking into account the coupling between horizontal motions. Such controllers are, until today, extensively used since they are structurally simple and do not require the complete modelling of environmental agents. Since the dynamics is non-linear with respect to the yaw heading of the barge, different linear models are obtained for each heading and the controller would have to be redesigned for all situations. Sometimes, new matrix weights must also be selected. A second and totally different approach is to take into account all environmental agents models during the design of the controller, taking profit of all information contained in such models. Furthermore, a non-linear controller can be used, avoiding the linearization problem, enabling the design to be done with the full non-linear model of the barge and environmental agents. However, since this methodology requires estimates of all environmental agents, namely current, wind and waves properties, the controller must also handle errors in these estimates. Environmental agents monitoring still presents practical difficulties evolving high levels of uncertainty. In other words, the controller must be robust with respect to the whole set of parameters variations. Furthermore, since complex hydrodynamic phenomena are involved in the wave and current-body interaction, the controller must also present robustness to modelling errors and guarantee performance and stability for models slightly different than that used in the design. epresenting this class of controllers, a robust version of the feedback linearization technique was developed using the sliding mode methodology originally proposed in [4]. Such methodology presents good robustness properties being adequate for models containing parametric uncertainties and modelling errors. In theory, the controller guarantees optimal performance with bounded model uncertainties; however, the control action presents highly oscillatory components (chattering), which is unsuitable for most applications. The control law can be modified to solve this problem [8], avoiding chattering and still guaranteeing a prescribed level of performance.

3 This paper describes the application of both controllers and the comparison between them, focusing on dynamic performance and energy consumption, number of adjusting parameters, implementation simplicity and robustness to modelling errors. The analyses showed that both controllers satisfy operational performance requirements. SM controller is, however, more appropriate, due to good robustness properties and fewer number of parameters to be adjusted. The LQ controller is simpler, but weight matrix adjustment is very time-consuming and must be redone whenever the barge heading changes. In the second section, all barge data are described, including the forces coefficient obtained in towing tank tests. The models used in the numerical simulator are also presented, including environmental agents, barge and propellers dynamics. In the third section, the application of the SM controller is described, with a theoretical explanation about this control methodology. The LQ controller is also briefly exposed. In the sequence, the comparison between the controllers is presented, with some illustrative simulations included. The conclusions are then summarized in the last section. Barge data and modelling BGL s main particulars are presented in Table. Table : BGL main data Length (L) Beam (B) Draft (T) Position of CG (x G ) Mass (M) Surge Added Mass (M ) * Sway Added Mass (M ) *.9 m.48 m 5.8 m -4.8m 777ton 77ton 8588ton Yaw Added Mass (M ) *.8. 7 ton.m Lateral Area (A L ) 5m Frontal Area (A F ) 4m * at low frequency For the S-lay operation, a constant force of 9kN applied by the pipe, acting backwards at the end of the stern ramp has been considered. A smaller force of kn has been considered for the J-lay mode analysis. Such a force is applied to an anchoring point at a not yet existing moon-pool. The direction of this latter force (launching direction) may change within a sector of ±9 o. Fig. 4 shows either applied forces. For simplicity, the forces are ideally considered as constant. (-9.7m;-.9m) S-Lay Launching Force (9tf) Stinger y (.m;7.m) J-Lay Launching Force (tf) Launching direction Figure 4: Launching operation. Pipeline applied forces (S-lay and J-lay) Thrusters position is shown in Figure 5. The system consists of three fore-body units and three stern ones. (-5 ;,5) #5 # (-,99 ; 9,5) #4 (-,99 ; -8,44) y # (5, ;,5) (59, ; -7,4) # (4,8 ; -,5) # Figure 5: Propeller Positions in meters (related to Y axis centered at CG) Figure shows the simulator block diagram. The full feedforward path is just present in the SM controller. For the LQ controller there is only a wind feedforward path, current and waves effects being treated as disturbances. Detailed descriptions of the controllers are given in sections 4 and 5. Set Point Controller Control Forces Filtered Motions Thruster Allocation Algorithm Estimation of Environmental Agents Intensities and Directions Control thrust and azimuthal angle in propellers Dynamics and properties of thurusters Wave Filter Actual thrust and azimuthal Waves Measured motions Environmental Agents Models Wind x Current Barge Dynamics Model Figure : Block diagram of simulator x

4 The wave filter used is a cascaded notch filter given in [4]. The simulator also models the propeller dynamics. The model also gives an estimate of total power consumed by each thruster. For simulation purposes, a series Ka propeller with a 9A nozzle, with m diameter and.m pitch has been used. The torque (K Q ), thrust (K T ) and nozzle (K TN ) coefficients are given in [5]. Drive system efficiency of each thruster is 8%, and the maximum power of each unit is 5kW (resulting, approximately, a maximum thrust of kn). Thruster control parameters were adjusted so that it takes approximately 5s for the thruster to raise from kn to kn. The thruster allocation algorithm adopted is based on a modified pseudoinverse technique by []. An azimuth filtering was implemented (to prevent the thruster from tear and wear), assuming maximum azimuth rotation velocity of 9 o /s. The following dynamic model gives horizontal motions of the barge: ( M + M) x ( M + M ) xx M x = F E O ( M + M ) x + M x + ( M + M) xx = F E O ( I + M ) x + M x + M x x = F. Z E O T T ; ;() where I z is the moment of inertia about the vertical axis; F E, F E, F E are surge, sway and yaw environmental loads (current, wind and waves),, F F are operation forces and moment due to F O O, O the pipe being launched and F T, FT, F T are forces and moment delivered by propulsion system. The variables x, x and x are the midship surge, sway and yaw absolute velocities (Figure 7). (sway) x Y x x = ψ (yaw) Figure 7: Coordinate systems (surge) Static current forces and moment are evaluated using the experimental coefficients (C, C Y and C M ) defined as: T C = C ρltu ; F C CY ρltu F = ; F C CM ρl TU = () were F C and F C are surge and sway current forces, F C is yaw moment, ρ is water density and U is the barge velocity, relative to the water. The coefficients, obtained in IPT towing tank, are presented in Figure 8. Current forces and moments associated with barge yaw rotation were evaluated using a cross-flow model presented in [7] Cy *Cx *Cm Current elated Heading (degrees) Figure 8: BGL Experimental Current Coefficients. Static wind forces were also evaluated using experimental coefficients (C V, C YV and C MV ) obtained in a captive wind test conducted, with the barge model superstructure turned upside down into the towing tank. Wind coefficients are defined in the standard way by: F V F V = CV ρa AFV ; ρ F V ρ = CYV a ALV ; = CMV a AL LV () where F V and F V are surge and sway components of the wind force, F V is the wind yaw moment; ρ a is the air density; V, the mean wind velocity. The corresponding coefficients are presented in Figure 9. Gust spectrum simulations consider Harris-DNV formula: S V 8ω ( ω ) = 4. C. V. + (4) V where ω is the frequency of wind oscillation, S v is the spectral density and C is a surface drag coefficient, used as.5 for moderate seas. 5

5 Dx and Dy (N/m) Cyv *Cmv Cxv Wind related heading (degrees) Figure 9: BGL Experimental Wind Coefficients.8E+5.E+5.4E+5.E+5.E+5 8.E+4.E+4 4.E+4.E+4.E+ -.E+4 -.E Wave Frequency (rad/s) Dx(o) Dy(o) -Dz(o).8E+.E+.4E+.E+.E+ 8.E+5.E+5 4.E+5.E+5.E+ Figure : BGL Wave Drift Coefficients for o wave heading angle A Jonswap sea wave spectrum was adopted and mean drift wave forces were evaluated using the drift coefficients obtained from a very well known and validated wave-body interaction computer code. Figure presents an illustrative example of such coefficients for a o wave heading angle. Second order slow drift forces are evaluated using []. Current-wave interaction effect on wave drift forces is estimated following []. First order wave motions are included in the model using the well known esponse Amplitude Operators (AO s). 4 Sliding Mode Controller Design The sliding mode controller is composed by an estimated feedback linearization action added to a term responsible to guarantee the prescribed performance and stability in face of limited errors. This term was originally defined in [4] as an onoff controller with amplitude proportional to the maximum modelling error. Due to implementation delays or numerical imprecision, a high oscillatory WAMIT Dz(N/m) control action results from the on-off controller. However, it can be shown that the use of a smooth transition avoids the occurance of that phenomenum [8], still guaranteeing the stability of the system despite a small performance loss. The sliding mode control was successfully applied to several non-linear systems, such as robot manipulators [9], OV s [5], ship track control [] and assisted dynamic positioning of moored vessels []. The later work describes in detail the application of such a controller to a system similar to the pipe-laying barge here considered. ewriting () in terms of the accelerations and velocities in the OY fixed coordinate system, being the reference point that lays in the center line of the barge, x ahead the midship section, one can easily obtain: f Y = f ψ f, din Y, din ψ, din ( ( ( ) F ) + C( ψ ) F ) F E E E O O O T T T (5) where f i,din are functions of the inertial properties of the barge and its velocity, C is a rotation matrix depending on the yaw angle and F i are the forces transferred to the reference point. Assuming that all terms in (5) are accurately known, the estimated feedback linearization controller that guarantees that the states follows the desired ones ( D, Y D, Ψ D ) is given by: F F F T T T = C fˆ fˆ f, din Y, din ψ, din ( ( ( ) Fˆ ) Fˆ ) ˆ F E E E ˆ ˆ ˆ M M M + C ~ ~ D λ λ ~ ~ Y λ λ D YY Y Y ψ λψψ~ λψ ψ~ D () where (~) denotes the tracking error and (^) the best estimate of the corresponding term. The control parameters λ are related to the bandwidth of the closed loop system. It is recommended to keep them smaller either than unmodelled resonant modes or than the inverse of phase lags of the system. In order to guarantee a prescribed tracking precision even with errors in the estimates, an extra term proportional to the maximum error is added to () given by:

6 F F F T T T K sat( s Φ ) = C K sat( s Φ ) (7) Y Y Y Kψ sat( s ψ Φψ ) where the variables s are true measures of tracking performance added to an integral action defined by ~ ~ ~ s = + λ + and analogously for Y and ψ. The gains K = (K, K Y, K Ψ ) T are obtained such that all elements of : ( max F E Fˆ E ) K C (8) are positive. It has been assumed that the inertial functions f i,din and the operational forces F O are known with sufficient accuracy compared to the environmental forces. The parameters Φ are responsible for the smoothness of the on-off action and must be tuned such that [8]: max( K ) i Φ i = (9a) λi Futhermore, it can be shown that the tracking precision is limited by: ~ i Φi max( Ki) < λ = (9b) λ i i Since λ i is limited by the unmodelled frequencies and time delays, it can be seen that for higher uncertainty about the system, higher is the expected tracking error. The calculation of maximum errors for environmental forces is done based on the uncertainty about the direction and intensity of the corresponding environmental agent. For example, supposing that the estimated current velocity is Uˆ and its direction is αˆ, with a maximum uncertainty of U and α respectively. The maximum error in current forces and moments is obtained using a Taylor expansion ( E = F Fˆ ): max E U ic ic ic [ EiC ( U, α) ] U + α + eic Fˆ ic Uˆ, αˆ ic ic E α Uˆ, αˆ () The later term is a non-parametric error also added to the controller. The derivatives can be evaluated either by analytical or numerical calculation. Expressions similar to () are used for winds and wave loads, and all together are used in (8). 5 LQ controller design Writing () for the reference point and eliminating quadratic terms one obtains: x M x = FT E x O () being the mass matrix M dependent only on well known mass properties. Besides, the velocities in the fixed coordinate system OY is related to the surge and sway velocities by: x Y = J ( ψ ) x, ψ x cosψ senψ J ( ψ ) = senψ cosψ () x = ψ as the state vector, and linearizing the model for heading angles near the operational one ( ψ ) one Defining ( x x x Y ) T obtains: M I x + J( oper ) x I = F T + oper which can be written in the state-space format: x = Ax + B. FT + D D () (4) being A and B directly obtained from () and D the disturbance vector containing the environmental and pipe laying forces. In order to include integral action to avoid steady offset errors, it can be defined the extended state space vector x ( x ) T ( D ). dt ( Y YD ). dt ( ψ D dt, ext ψ = ). (5) And the new state-space equation is defined by the following extended matrixes: A A B ext = I B ext = ()

7 The control forces are then given by the state feedback law FT = Kx such that the gain matrix K is obtained from the associated steadystate iccati equation: A T ext K = S + SA B ext T ext S SB ext B T ext S + Q = (7) where Q is the state and is the control weight matrixes. Case studies and comparison between controllers A common Brazilian Campos Basin environmental, condition shown in Figure, is considered in the present analysis. H s is the significant wave height and T p is the peak period. Current (8 ) U=.m/s 7 Waves (9 ) Hs=.5m Tp=8.45s Y 9 Wind ( ) V=m/s 8 Figure : Environmental conditions The main task of the barge DPS is to follow a prescribed path determined by the pipe being launched. For the S-lay operation, Figure shows a typical path that must be followed with a constant tangential velocity of.5m/s. =m Figure : Typical reference path The SM controller is designed assuming that all environmental agents are collinear, with a direction of 9 o. Furthermore, all estimated values used in the design, as well as the modelling errors imposed to the simulation, are shown in Table. It is important to mention that the controller was designed for conditions very different from the real ones, as a robustness test. Furthermore, a bandwidth factor (λ) of.4 is used for all motions. Table : SM design parameters and modelling errors imposed in the simulation Parameter eal Value Estimated Value( * ) Max. Variation( * ) Current Velocity U=.m/s Û=.m/s U=.m/s Current Dir. α=8 o ˆ =9 o α= o Wave Sig. Height H s =.5m S=.m H S =.m Wave Dir. α W =9 o ˆ W =9 o α W = o Wave Peak Per. Τ P =8.45s Tˆ = P 8.s P Mean Wind Dir. α Wi = o ˆ Wi =9 o α Wi = o Current/Wave/ +% % Wind mod. error Wind Velocity Measuring Error % % ( * )considered in the design The resulting path of the barge is presented in Figure, and tracking errors in Figure 4. Since the admissible launching error is 7,5m, the controller fully satisfies performance requirement, also keeping the heading error smaller than.5 o. Y (m) (m) Posição Pto eferência Figure : esulted path using SM controller Y error(m) Ψ error(degrees) 4 Y (metros) (metros) beginning of curve Time(s) end o f curve Figure 4: Tracking and heading errors: SM control

8 The LQ controller was also tested under the same environmental conditions and modeling errors shown in Table. The weight matrixes were adjusted to reach an acceptable performance. The simulation is shown in Figures 5 and. The model was linearized about o heading. The performance is worse than the SM controller, though still attending the required maximum 7,5m deviation. The heading error is kept smaller than o after the transient. It must be stressed that the weight matrixes determines the performance, and possibly better results could be obtained by other choices not tested here. For both controllers the mean total power consumption was approximately 5kW with a peak consumption of 7kW. Some differences between both control philosophies are exemplified by this example. It must be stressed again that both of then satisfied the performance requirements for the pipe-laying operation. Y (m) (m) Posição Pto eferência Figure 5: esulted path using LQ controller Y error(m) Ψ error(degrees) 4 Y (metros) (metros) beginning of curve Time(s) end o f curve Figure : Tracking and heading errors: LQ control The SM controller requires estimates of all environmental conditions and boundaries in the errors on these estimations. Additionally, it also requires the closed loop system bandwidth, which can be evaluated by the desired response of the final system. Since the controller knows all physical phenomena involved in the process, because all the models of environmental forces and dynamics are included, it does not require any other adjustment. The performance is then guaranteed for all possible maneuvers if the environmental conditions remain inside the predictions. Furthermore, the controller can be continually tuned in order to accommodate slow variations of waves, wind and current. The only requirements are simple estimations of some parameters, namely Hˆ, αˆ, Tˆ, Uˆ, αˆ, αˆ. S W P Wi An apparent disadvantage of this class of controller is that it requires estimations of the environmental conditions, what may be expansive and may imply practical difficulties. Due to this problem, the SM controller has been designed to handle variations on those estimations, in such a way that even poor estimations are acceptable. The case study, e.g., showed that the controller is able to handle a 5% variation in the significant wave height. Furthermore, despite the well developed wind sensing technology, some new techniques are being developed to measure current and waves. For example, vertical motions of the ship can be used to estimate the wave incident spectrum, with maximum errors which can be easily handled by a robust controller design []. Another apparent disadvantage seems to be the complexity of a model-based controller such the SM. However, nowadays, since oceanic systems require sampling time of approximately s, this is not a real problem even with the use of common personal computers. Conversely, the LQ controller represents a class of blind controllers which consider all environmental forces (except wind) as disturbances in the control loop. As can be seen, the performance requirements can be attended even without any information about environmental conditions. The tuning process is, however, very important, and from different matrixes would result very different performances. Exhaustive simulations must be done to select the best weight matrixes guaranteeing that performance

9 requirements are satisfied for the possible environmental conditions. The second important point that must be analyzed is related to the linearization necessary to the design of LQ controller. For the present, the controller so obtained for an operation heading of o guarantees stability for heading angles between 4 o and 4 o. The stability boundary can be easily obtained by the calculation of the eigenvalues of the closed loop matrix of system (4) controlled by (7). Figure 7 illustrates the possible headings that the barge may reach safely with the LQ controller. +4 o Stable Headings for the LQ controller designed with the model linearized about the actual state of the ship. However, such controller would be extremely complex and the stability of such gain scheduled system would be questionable. Y error(m) Ψ error(degrees) Time(s) Figure 8: ectilinear 5 o path: LQ control 4 Figure 7: Stable headings for the LQ controller Of course, the controller performance gets worse if the heading angle approximates the boundaries values. The SM controller, conversely, does not require the linearization of the model, what guarantees stability and good performance for all headings. For the sake of illustration, a rectilinear 5 o path is considered. The simulations are conducted under the same environmental conditions and modeling errors used in the first case. The initial heading error is 5 o and both controllers were applied. As expected, Figure 8 shows that the response of the system under LQ controller is extremely oscillatory, since the heading is close to the stability boundary. The performance of the system gets worse, taking approximately 8s to decrease the oscillation to an acceptable level. The SM controller, for its turn, reaches the desired path and heading after a short transient and without oscillations (Figure 9), keeping the overall good performance presented in the first simulation. If a linear controller is used, a full stability analysis must be done, defining the allowable headings the system may reach. A possible approach would be the use of several linear controllers, each one -4o Y error(m) Ψ error(degrees) Time(s) Figure 9: ectilinear 5 o path: SM control 7 Conclusions In the present work the control of a pipe laying barge has been analyzed, focusing on the comparison between two different philosophies commonly used in DP systems. A numerical simulator was developed in a previous work and used here. Current and wind forces are evaluated using towing tank tests and additional effects are included using fully validated models. Wave effects are considered using an wave-body interaction software. The simulator includes the azimuthal propeller dynamics and a thrust allocation algorithm. The first class of controllers tested is a linear and non-model based control, commonly used since early times in DPS applications, designed by LQ theory. Second category is composed by non-linear model-based control, here represented by a Sliding Mode controller.

10 Despite the simplicity of the LQ control and the fact that it does not require any information about the environmental conditions, the weight matrixes selection is a very time-consuming task, and determines the overall performance of the controller. Exhaustive simulations are required to verify if the performance requirements are satisfied for all environmental conditions and desired reference paths since the controller design cannot guarantee itself the performance for all those conditions. Furthermore, a stability analysis must be carried out, and several controllers must be used for different headings since nonlinear effects are disregarded in the design. The SM control is, as far implementation is concerned, more complex, because it encloses a model of the system and environmental agents and requires estimates of environmental conditions. However, due to robustness properties, the controller guarantees stability and performance in the presence of estimation and modeling errors, bounded by prescribed maximum values. Therefore, simple systems for environmental conditions monitoring (or even real time estimators based on ship motion) can be used, since the controller can handle large errors on these estimations. The complexity of the controller is not a problem for the computational capacity of modern computers added to the fact that the ocean systems dynamics are extremely slow. Acknowledgements This work has been supported by Petrobras, State of São Paulo esearch Foundation (FAPESP Proc.No 98/98-) and CNPq/CTPETO (process no. 4995/). The authors thank Mr.. Pesce for helping with simulations and diagrams. eferences [] J.A.P.Aranha, A.C. Fernandes, On the secondorder slow drift force spectrum, Applied Ocean es., Vol.7, pp.-, (995). [] J.A.P. Aranha, Second order horizontal steady forces and moment on a floating body with small forward speed, J. Fluid Mec., Vol, (99). [] D. Bray, Dynamic Positioning, The Oilfield Seamanship Series, Vol.9, Oilfield Publications Ltd. (OPL), (998). [4] T.I., Fossen, Guidance and Control of Ocean Vehicles, John Wiley and Sons, Ltd., (994). [5] E.V. Lewis, Principles of Naval Architecture, SNAME, Jersey City, (988). [] F.A., Papoulias,A.J., Healey, Path control of surface ships using sliding modes, J. Ship es., Vol., No., pp4-5, (99). [7] A.N. Simos, et al., A quasi-explicit hydrodynamic model for the dynamic analysis of a moored FPSO under current action, J. of Ship es., Vol.45, No.4, pp89-, (). [8] J.J.E., Slotine, Sliding controller design for non-linear systems, Int. J. Control, Vol.4, No., pp4-44, (984). [9] J.J.E., Slotine, The robust control of robot manipulators, Int. J. obotics esearch, Vol. 4, No., pp49-4,(985). [] O.J. Sørdalen, Optimal Thrust Allocation for Marine Vessels, Control Engineering Practice, Vol.5, No.9, pp-, (997). [] E.A. Tannuri, D.C. Donha, C.P. Pesce, Dynamic positioning of a turret moored FPSO using sliding mode control, Int. J. obust and Nonlinear Cont., Vol.,, pp9-5, (). [] E.A. Tannuri, C.P. Pesce, D.C. Donha, Assisted dynamic positioning system for a FPSO based on minimization of a cost function, CAMS Proceedings, Scotland, (). [] E.A. Tannuri et al., Dynamic Positioning of a Pipeline Launching Barge, ISOPE Conference, Kyushu, Japan, (). [4] V.I., Utkin, Sliding Modes and their application to variable structure systems, MI Publishers, Moscow, (978). [5] D.., Yoerger, et al., Supervisory Control System for the JASON OV, IEEE J. on Oceanic Eng., Vol. OE-, No., pp9-4, (98).